Semelhança de triângulos através da resolução de problemas: uma análise do pensamento geométrico

Abstract

This research aims to analyze the Geometric Thinking mobilized by students in Middle School during the implementation of Teaching-Learning-Evaluation Methodology of Mathematics through Problem Solving, when solving generating problems, exploring the object of knowledge: triangle similarity. The research qualitative and the participants were 23 students of the last grade of Middle School from a social school in the city of Londrina/Paraná. The research instruments were five generating problems on the theme, implemented during ten hours/class. The data were collected through written records by the students and audio recordings transcribed from the moment of problem solving, from observations and records in the field diary of the teacher-researcher. These data were categorized in two moments (the ones dedicated to solving the problem and to the plenary) and analyzed according to the theoretical framework presented and the objective of the research. Groups G1 and G4 were analyzed. The first group started at Van Hiele level 0 (visualization – classification of figures), went through level 1 (analysis – properties of figures) and finished the sequence of problems at level 2 (informal deduction – relationship between the properties of figures). G4 started at level 2 (informal deduction – relationship between the properties of figures) and finished at level 3 (deduction – properties deduction). The results indicated that Geometry, when explored through Problem Solving, allowed students to learn a new approach to learning Mathematics and provided opportunities for the development of Geometric Thinking: regarding the levels of Geometric Thinking observed and characterized, it was evident that, during problem 3, the G1 students demonstrated characteristics of Van Hiele level 0 (visualization – when they grouped the shapes according to their appearance) and the G4 students demonstrated level 2 (abstraction and informal deduction of the “if-then” type), in addition to the previous ones. In problem 4, G1 also demonstrated characteristics of level 0, and G4 demonstrated levels 0 and 1 of Geometric Thinking. Finally, in problem 5, G1 and G4 demonstrated level 2 and earlier, and some G4 students already showed signs of advancing to level 3. An Educational Product called “Methodology for Teaching-Learning-Assessment of Geometry through Problem Solving” was produced during the research, containing the five problems worked on and validated, as well as guidelines and new suggestions for problems for teachers who want to explore the theme.